3.2 Stationary states and scattering in one dimension
The key idea of the stationary-state approach is to avoid treating individual particles, and to consider instead the scattering of a steady intense beam of particles, each particle having the same energy E0. It is not possible to predict the exact behaviour of any individual particle but, if the incident beam is sufficiently intense, the result of the scattering will be reflected and transmitted beams with steady intensities that are determined by the reflection and transmission coefficients we are aiming to evaluate. Provided we consider the beams as a whole, nothing in this arrangement depends on time. A snapshot of the set-up taken at one time would be identical to a similar snapshot taken at another time. In contrast to the wave-packet approach, there are no moving ‘blobs of probability density’, so the whole process can be described in terms of stationary states.
For a one-dimensional beam, we define the intensity j to be the number of beam particles that pass a given point per unit time. We also define the linear number density n of the beam to be the number of beam particles per unit length. Then, thinking in classical terms for a moment, if all the particles in a beam have the same speed v, the beam intensity is given by j = vn. Specialising this relationship to the incident, reflected and transmitted beams, we have
In the stationary-state approach, the reflection and transmission coefficients can be expressed in terms of beam intensity ratios, as follows:
If all the incident particles are scattered, and no particles are created or destroyed, it must be the case that jinc = jref + jtrans. Dividing both sides by jinc and rearranging gives R + T = 1, as expected from our earlier discussions.
We now need to relate these steady beam intensities to stationary-state solutions of the relevant Schrödinger equation. This requires some care, since Schrödinger's equation is normally used to describe individual particles, rather than beams of particles. To make the steps in the analysis as clear as possible, we shall begin by considering a particularly simple kind of one-dimensional scattering target.